"After throwing together several of our own observations [on the structure of vessels, the utility of British forest trees for naval timber, practical matters of nursing, planting, training, pruning etc.], we bethought ourselves of examining into the ideas and experience of recent writers on the same subject." (p. 138)

"Since writing the above, we have looked over some experiments by Messrs Barlow, Beaufoy, Couch, and others, on the strength of timber." (p. 221, see Barlow, 1826 and citations therein)

"After the preceding parts of this volume had gone to press, we received a copy of Cruickshank's Practical Planter." (p. 309)

Horizontal lines
While the thematic breaks in the contents tell us something about the miscellany topics of the book (see first paragraph) and Matthew's asides tell us something about his discontinuous writing and thinking (see second paragraph), there is a third intriguing feature of the book—horizontal lines interrupting the main text (horizontal lines that merely underline headings or sub-titles will be ignored). These text-interrupting horizontal lines occur at the pages 135, 221, 294, 358, 381, 390:

The first thing to note about these horizontal lines is that they do not consistently mark thematic breaks, that is, some thematic breaks are accompanied by such a marker (e.g., pp. 221, 381, 390), but not all thematic breaks in the book are marked that way. For example, the complete thematic break at page 388 is not marked by a horizontal line. Here, Matthew ends his important addendum that is still relevant for the history of evolutionary ideas, today, and begins with a corrigendum about his figure at page 27 (part d of that figure) and about his directions for forming larch roots into knees. However, this break is not marked by a horizontal line or another conspicuous visual element (no heading or anything either). Instead, one line is merely left blank and the text simply continues with one of Matthew's asides (see middle page in the following image):

Equally unmarked thematic breaks occur elsewhere in the book as well.
Second, the horizontal line at page 135 does not mark a thematic break, but rather a thematic elaboration. That is, after emphasizing the importance of the marine for the superiority of the British economy at pages 130-134, a horizontal line occurs at the top of page 135 and Matthew continues to elaborate that this importance of the marine for the British economy should be reflected in a direct political representation of the marine in the British parliament. [This elaboration contains another elaboration in the form of a footnote at page 135 referring to the end-note E in the appendix. This Note E (p. 376-7), in turn, rants against ships being taxed by their length and widest breadth rather than by their tonnage leading to ill designed ships being deep and bulky rather than streamlined.] He ends this elaboration about the representation of the marine's interests by stating that, in the absence of such a direct representation in the parliament, the hopes of Britain rest with its sailor kings:

"The existence of Britain depends upon her Marine, and the king should always be bred a sailor—the heir-apparent and presumptive being always sent to sea. In the case of a female, if she did not take kindly to the sea-service, a dispensation might be allowed, on her marrying a sailor, and the foolish law prohibiting our Royal Family from marrying a Briton be put aside." (p. 137).

Likewise, the horizontal line at page 294 marks the beginning of an elaboration of another sort. Here, Matthew switches from criticizing Steuart's Planter's Guide to criticizing an author (Loudon) that Steuart has quoted at length in one of his end-notes (see here, here and here).
Again, the horizontal lines are neither consistently associated with the asides by Matthew quoted above. While some of the horizontal lines occur atop of an aside from Matthew (e.g., at pp. 221, 390), not all of the asides are accompanied thus (e.g., pp. 309, 388). Apparently, the function of these horizontal lines is not one for the reader.

The significance of horizontal lines in Matthew (1831)
Given that Matthew was a re-iterative writer adding passages in proof or even while the book was already in press, the horizontal lines may have been marks for the publishers (Adam Black, Edinburgh & Longman et al., London) or the printers (Neill & Co., Edinburgh) telling them where to insert the late additions of Matthew. If this was true, then the thematic breaks would reveal something about Matthew's thinking and his asides about his discontinuous writing, but the horizontal lines would indicate the cleavage sites, where the publishers or printers inserted Matthew's late additions. Some of them were inserted singly others together as indicated by thematic breaks or asides. Testing this hypothesis about the function of the horizontal lines by reverse engineering yielded the following surprise.
Apparently, the elaboration on Steuart's long quote of Loudon (starting after the horizontal line at the bottom of p. 294), Matthew's spliced re-quote (see here, here and here) and his criticism of this Loudon re-quote (extending from pp. 295-308) got inserted along with the whole part criticizing Cruickshank's Practical Planter (as indicated by the aside at p. 309). The amazing thing that happens, when the horizontal lines in Matthew (1831, pp. 294 + 358) are taken as the joints for dissecting and splicing out the intron (the Loudon loop + the Cruickshank part) [Best use the Project Gutenberg html-version for this job.], is that the result is a rather well composed neat ending for the book:

"We begin to think, from our disposition to ramble from the Allanton system [meaning Steuart, who lived at Allanton House and had proposed a system of transplanting large trees in whole], that we tire of Sir {294} Henry; and we believe, should he follow us thus far, that he will be tired of us. On looking back on what we have written, we are almost disposed to accuse ourselves of being splenetic; but the truth is, we regard the whole art as very unimportant, if not positively pernicious, at least in the way in which it has been exemplified by Sir Henry, as a throwing away of valuable labour to no purpose, if it ought not indeed to be considered as a mere pander to luxury and caprice. We have no sympathy with the aristocratical object of the book, and as little with the aristocratical tone in which it has been bepraised by Sir Walter Scott. We should also have no greater pleasure in the discovery of a royal road to virtue than we should have to the discovery of one to science,—the four cardinal virtues being, as every body knows, writing books, building houses, and raising trees and children, but we should hope, neither by proxy, nor by the Allanton System. While, however, we thus state our opinions with freedom, we do not hesitate to add, that Sir Henry’s volume has afforded us more information, or, at least, more materials for reflection, than any other of the works which we have brought under the notice of our readers.

{text between horizontal lines at pages 294 and 358 spliced out}

We have now brought before the reader a pretty fair picture of the Forestry of the present day. Some may wonder that the written science of arboriculture should be so imperfect and inaccurate; but the knowledge of the art, and the power of communicating that knowledge, are of so different a {359} character, it not unfrequently happens, that those write who cannot act, and those who can, are incompetent to write—sometimes unwilling; besides, correct opinions on this subject, as on most others, are only just beginning to be formed. We have endeavoured to assist in disentangling the correct from the erroneous. It is impossible for the most wary always to avoid misconception of facts, but man merits the name of rational only, when he evinces a readiness to break from those misconceptions, to which the narrow-minded, the proud, the vain, and the creature of habit and instinct, cling so obstinately. As a friend, we have stood on no ceremony with our brother arboriculturists. We have laid ourselves open to their criticism, and we hope they will shew as little ceremony with us."

This is a crisp, apt and fluid ending of the main text of the book in place of the distinctive halting flow and discontinuity between sections (ignoring for the moment the equally discontinuous appendix with a list of end-notes followed by the evolutionary addendum, a corrigendum, a colophon and, finally, the errata).

External evidence from other contemporary books

Books with no text-interrupting horizontal lines
The one long argument On the Origin of Species by Darwin (1859) has not one horizontal line interrupting the main text. Only two underline the title of the book and the title of the contents page (see image below). Nor do any such lines crop up in later editions, where Darwin added or edited his earlier versions.

In order to exclude the possibility that horizontal lines became unpopular after 1831, but cluttered all books up to that date, take a look at Volume I of Erasmus Darwin's Zoonomia, first published in 1794 by J. Johnson and printed in St. Paul's Church-yard in London. The title page has two horizontal lines that separate a stanza from Virgil, which is inserted between the author and publisher information. Another pair of horizontal lines separates an entreaty, that the reader may endure a string of definitions necessary at the very beginning of the book. This is inserted between the summary and the text of that chapter (see image). No text-interrupting horizontal lines occur in the remainder.

Another book from about 30 years before 1831 is the English translation of George Cuvier's Lectures on Comparative Anatomy (1802, vol. I). Here the translators apparently economized on pages by separating chapters through horizontal lines instead of page breaks. This book comes in parts called lectures and each lecture falls into chapters called articles. When one article ended in about the upper two third of a page, a horizontal line would separate it from the next article continuing on the same page. If, however, an article ended close to a page break, no ink or space would be wasted for a horizontal line and the next article simply start at the next page. Lectures were not separated thus, but a new lecture begins at a new page regardless of the space left empty. Again, horizontal lines do not wantonly occur in the middle of chapters/articles.
A book cited by Matthew (1831, p. 221) in one of his asides has no text-interrupting horizontal lines either (ignoring one that separate a footnote and others that separate adverts appended after the end of the book as well as the usual underlines of titles and sub-headings). This is Peter Barlow (1826). An Essay on the Strength and Stress of Timber, founded upon experiments performed at the Royal Military Academy, third edition. Printed for J. Taylor at the Architectural Library in London.
Finally, Matthew's own Emigration Fields (1839) has no text-interrupting horizontal lines and no aside, either, of the form: "Since this work went to press ..." This is despite the fact that the publishers forced Matthew to augment his first draft (see Matthew 1839, p. v), which was exclusively on New Zealand, by similar chapters on North America, Mexico and Australia.
Adam Black was now joined by his brother Charles as the Edinburgh publishers, and his London publishers had exchanged Rees (in Longman, Rees, Orme, Brown, and Green) by Longmans (in Longman, Orme, Brown, Green and Longmans). The printers were, again, Neill & Co. in Edinburgh. This suggests to take the discontinuous writing style and high frequency of text-interrupting horizontal lines as a special feature of the book On Naval Timber rather than attributing it to the author, the publishers, or the printers. Matthew's publishers and printers had probably gone through a steep learning curve in dealing with him as an author the first time and simply would not move a thumb or cogwheel before the fat lady had sung the second time.

These examples just serve to show that well written books did not need such bric-a-brac, neither before nor after 1831, and that the parties involved in 19th century book production were not in the habit of wantonly sprinkling horizontal lines as decorations all over their texts.

Books with text-interrupting horizontal lines
Like Matthew's book, Cruickshank's Practical Planter(1830) has been printed by Neill & Co., though his publishers were William Blackwood (Edinburgh) and T. Cadell (London). Matthew may even have learned about it from the printers given how he added his critique of Cruickshank while the rest of the book was already in press and given his aside saying that "we received a copy" (Matthew 1831, p. 309). Cruickshank contains but one horizontal line in the main text at page 53, and it marks a paragraph that has definitely been added as an afterthought. The whole chapter before is just an enumeration of different trees with a paragraph or two of superficial observations on its habitus, timber etc. But at the end of that chapter, he bethought himself to say something about the special mode of propagation (not by seed) of some of the trees, in particular: the lime, willows, poplars by layers or cuttings respectively.

Steuart's Planter's Guide (1828) was, again, published by William Blackwood (Edinburgh) and T. Cadell (London) with no separate printer being speciefied (Blackwood did inhouse printing, pers. comm., Julian Derry). Again, the whole book contains but one text-interrupting horizontal line at page 274, but this one is tricky.

The text following that marker reads like a direct address of the readers breaking the fourth wall. One could conclude, at first glance, that the horizontal line signals this to the reader and has no function in showing the publishers or printers, where to insert a late addition. However, on perusing the book more extensively, one soon realizes that Steuart's style is that of soliloqui. In speaking with himself (or his other self), he points to "those" trees or "those," to "those" readers or "those" and observes their different needs. He does address the readers in this way at various places (e.g., pp. 4, 230, 234), but none of these other direct addresses got marked by horizontal lines. It is therefore likely that this horizontal line at page 274 also marks a late addition by Steuart.